Question

In Fourier series expansion, bn will zero for _______ function and anwill be zero for _______ function.​

Answer 1

Answer:

Even function

Odd function

Step-by-step explanation:

Answer 2

Concept: A periodic function, f(x), is expanded into the Fourier series as an infinite sum of sines and cosines. The orthogonality between the sine and cosine functions is used in Fourier series. Harmonic analysis, the computation and study of Fourier series, is very helpful for breaking down any arbitrary periodic function into a set of straightforward terms that can be plugged in, solved separately, and then recombined to get the answer to the original problem or an approximation to it to whatever accuracy is desired or practical. The Wolfram Language's implementations of the Fourier cosine coefficient ($a_{n}$) and sine coefficient ($b_{n}$) are Fourier Cos Coefficient [expr, t, n] and Fourier Sin Coefficient[expr, t, n], respectively.

Given: Fourier series expansion

           $b_{n}$ = 0

           $a_{n}$ = 0

To find: the function for which $b_{n}$ will be equal to 0

             the function for which $a_{n}$ will be equal to 0

Solution:

f(x) sin(n x) is odd if a function is even such that f(x) = f(-x).

[Since sin(n x) is odd and an even function multiplied by an odd function is an odd function, this conclusion follows.]

Therefore, for any n, $b_{n}$ = 0.

Likewise, f(x) cos(n x) is odd if a function is odd such that f(x) = -f(-x).

[Since cos(n x) is even and an odd function multiplied by an even function is an odd function, this follows.]

As a result, for any n, $a_{n}$ = 0.

Hence, In Fourier series expansion, $b_{n}$ will be zero for odd function and $a_{n}$ will be zero for even function.

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